Existence of non-Abelian vortices in a coupled 4D-2D quantum field theory

Vortices produce locally concentrated field configurations and are solutions to the nonlinear partial differential equations systems of complicated structures. In this paper, we establish the existence and uniqueness for solutions of the gauged non-Abelian vortices in a coupled 4D-2D quantum field theory by researching the nonlinear elliptic equations systems with exponential terms in $\mathbb{R}^{2}$ using the calculus of variations. In addition, we obtain the asymptotic behavior of the solutions at infinity and the quantized integrals in $\mathbb{R}^{2}$.